Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group

Let $M$ be a nilmanifold with a fundamental group which is free $2$-step nilpotent on at least $4$ generators. We will show that for any nonnegative integer n there exists a self-diffeomorphism $h_n$ of $M$ such that hn has exactly $n$ fixed points and any self-map $f$ of $M$ which is homotopic to $h_n$ has at least $n$ fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes.

Keywords

fixed point theory, Nielsen number, Reidemeister number, free nilpotent group, nilmanifold

2010 Mathematics Subject Classification

Primary 55M20. Secondary 20F18, 37C25.

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The authors’ research was supported by long-term structural funding from the Methusalem grant of the Flemish Government.

Received 27 October 2017

Accepted 7 May 2019

Published 21 August 2020